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Multi-granulation Fuzzy Rough Set Model on Tolerance Relations
Weihua Xu, Qiaorong Wang, and Xiantao Zhang
Abstract— Based on the analysis of rough set model on atolerance relation and considering of the theory of fuzzy roughset, two types of new generalized fuzzy rough set models areconstructed, which are multi-granulation fuzzy rough sets ontolerance relations. It follows the research on the propertiesof the lower and upper approximations of the new multi-granulation fuzzy rough set models on tolerance relations. Thefuzzy rough set model and rough set model on a tolerance rela-tion are special cases of the new one from the perspective of theconsidered concepts and granular computing. The relationshipsamong the fuzzy rough set model, the first MGFRS and thesecond MGFRS all on tolerance relations are investigated.
I. INTRODUCTION
ROUGH set theory, proposed by Pawlak [1], is a theory
for the research of uncertainty management in a wide
variety of applications related to artificial intelligence. The
theory has been applied successfully in the fields of pattern
recognition, medical diagnosis, data mining, conflict analysis,
algebra, which related an amount of imprecise, vague and
uncertain information. In recent years, the rough set theory
has generated a great deal of interest among more and more
researchers. The generalization of classical rough set model
is one of the most important study spotlights.
As we know, the classification of objects in the classical
approximation space is based on the approximation classifi-
cation of equivalence relations. This kind of classification is
very restrictive. So it is necessary to relax the equivalence re-
lations to tolerance ones for the need of some practical issues.
On the basis of this point, some researchers extended the
classical approximation space to tolerance one and discussed
the reduction approach of objects sets [2]-[4]. Besides, rough
set theory is generalized by combining with other theories
that deal with uncertainty knowledge such as fuzzy set. It has
been acknowledged by different studies that fuzzy set theory
and rough set theory are complementary because of handling
different kinds of uncertainty. Dubois and Prade proposed
concepts of rough fuzzy sets and fuzzy rough sets based on
approximations of fuzzy sets by crisp approximations spaces,
and crisp sets by fuzzy approximation spaces, respectively
[5]. Yao proposed a unified model for both rough fuzzy sets
and fuzzy rough sets based on the analysis of level sets of
fuzzy sets in [6]. A rough fuzzy set is a pair of fuzzy sets
resulting from the approximation of a fuzzy set in a crisp
approximation space, and a fuzzy rough set is a pair of fuzzy
This paper was received on 29 June 2011. And this work is supportedby Postdoctoral Science Foundation of China (No.20100481331) and Na-tional Natural Science Foundation of China (No. 61105041, 71071124 and11001227).
Weihua Xu, Qiaorong Wang and Xiantao Zhang are with the Schoolof Mathematics and Statistics, Chongqing University of Technology,Chongqing, P. R. China (email: chxuwh@gmail.com).
sets resulting from the approximation of a crisp set in a fuzzy
approximation space.
Rough set theory was also discussed with the point view
of granular computing. Information granules refers to pieces,
classes and groups divided in accordance with characteristics
and performances of complex information in the process
of human understanding, reasoning and decision-making.
Zadeh firstly proposed the concept of granular computing and
discussed issues of fuzzy information granulation in 1979
[7]. In the point view of granulation computing, the classical
Pawlak rough set is based on a single granulation induced
from an indiscernibility relation. And an equivalence relation
on the universe can be regarded as a granulation. For the need
of some practical issues, Qian and Xu extended the Pawlak
rough set to multi-granulation rough set models where the
approximation operators are defined by multiple equivalence
relations on the universe [8]-[
Associated tolerant rough set with the theory of fuzzy
rough set with granular computing point view, we will pro-
pose two types of multi-granulation fuzzy rough set models
on tolerance relations. The main objective of this paper is to
extend J. Jarinen’s tolerance rough set model determined by
single tolerance relation to multi-granulation fuzzy rough sets
where set approximations are defined by multiple tolerance
relations. The rest of this paper is organized as follows. Some
preliminary concepts of tolerance rough set theory and fuzzy
rough sets theory are showed in Section II. In Section III,
for a fuzzy target information system, based on multiple
ordinary tolerance relations, two types of multi-granulation
fuzzy rough approximation operators of a fuzzy concept are
constructed and a number of important properties of them are
discussed in detail. Especially, one can find that the definition
of lower and upper approximation operators proposed in this
paper are the generalized model of other formats, not only
from the aspects of the considered concepts but also from the
perspective of granulation. Then it follows the comparison
and relations among the properties of the two types of multi-
granulation fuzzy rough sets on tolerance relations and fuzzy
rough set. And finally, the paper is concluded by a summary
in Section IV.
II. PRELIMINARIES
In this section, we will first review some basic concepts
and notions in the theory of rough set on tolerance relation
and fuzzy rough set on the basis of equivalence relations.
More can be found in Ref. [2].
A. Rough Set On A Tolerance Relation
The notion of information system provides a convenient
tool for the representation of objects in terms of their attribute
Fourth International Workshop on Advanced Computational Intelligence Wuhan, Hubei, China; October 19-21, 2011
978-1-61284-375-9/11/$26.00 @2011 IEEE
relations on the universe [8].
357
values.
A tolerance information system [4] is an ordered triple I =(U,AT, τ), where U is the non-empty finite set of objects
known as universe; A is the non-empty finite set of attributes.
τ is the mapping from powerset AT into the family set Rof tolerance relations satisfying reflexivity and symmetry on
universe U .
Let I = (U,AT, τ) be a tolerance information system. The
lower approximation and the upper approximation of a set
X ⊆ U on a tolerance relation R with respect to A ⊆ ATare respectively defined by
RA(X) = {x ∈ U |RA(x) ⊆ X},RA(X) = {x ∈ U |RA(x) ∩ X �= ∅}. (1)
where RB(x) is the tolerance class of x with respect to the
tolerance relation RB . Notice that a tolerance relation can
construct a covering instead of a partition of the universe
U . The set BndR(X) = RA(X) − RA(X) is called the
boundary of X .
The set RA(X) consists of elements which surely belong
to X in view of the knowledge provided by R, while
RA(X) consists of elements which possibly belong to X .
The boundary is the actual area of uncertainty. It consists of
elements whose membership in X can not be decided when
R-related objects can not be distinguished from each other.
The properties of the lower approximation and the upper
approximation of sets with respect to a tolerance relation RA
are as follows, if X, Y ⊆ U and ∼ X is the complement of
X .
(1)RA(X) ⊆ X ⊆ RA(X);(2)RA(∅) = RA(∅) = ∅, RA(U) = RA(U) = U ;(3) ∼ RA(X) = RA(∼ X),∼ RA(X) = RA(∼ X);(4)BndR(X) = BndR(∼ X);(5)X ⊆ Y ⇒ RA(X) ⊆ RA(Y ) and RA(X) ⊆ RA(Y ).
B. Fuzzy Set And Fuzzy Rough Set
We will first introduce some basic concepts of fuzzy set.
Let U be a finite and non-empty set called universe. A fuzzy
set A is a mapping from U into the unit interval [0, 1] : μ :→ [0, 1], where each x ∈ U is the membership degree of
x in A. Practically, we may consider U as a set of objects
of concern and crisp subset of U represents a “non-vague”
concept imposed on objects in U . Then a fuzzy set A of Uis thought of as a mathematical representation of “vague”
concept described linguistically. The set of all the fuzzy sets
defined on U is denoted by F (U).Let A be a fuzzy set on U , for any α ∈ [0, 1], if denote
Aα = {x ∈ U | A(x) ≥ α}.then Aα is the α-cut set of A.
Let U be the universe, R be an equivalence relation, for
a fuzzy set A on U , if take
R(A)(x) = ∧{A(y)|y ∈ [x]R},R(A)(x) = ∨{A(y)|y ∈ [x]R}, (2)
then R(A) and R(A) are called the lower and upper approx-
imation of the fuzzy set A with respect to the relation R,
where “∧” means “min” and “∨” means “max” and [x]Ris the equivalence class of x with respect to equivalence
relation R. A is a fuzzy definable set if and only if A satisfies
R(A) = R(A). Otherwise, A is called a fuzzy rough set.
Let I = (U,AT, F ) be an information system, Dj : U →[0, 1](j ≤ r), if denote
D = {Dj | j ≤ r},then (U,AT, F, D) is a fuzzy target information system.
In a fuzzy target information system, we can defined the
approximation operators with respect the decision attribute
D similarly.
Because of the limitation of the paper length, the properties
of the above set approximation and measures have been
showed in the reference [10].
III. TWO TYPES OF MULTI-GRANULATION FUZZY ROUGH
SETS ON TOLERANCE RELATIONS
In this section, we will make researches about multi-
granulation fuzzy rough set on tolerance relations which are
on the problem of the rough approximations of a fuzzy set
on multiple tolerance relations.
At first, we will propose a fuzzy rough set model (in brief
FRS) on a tolerance relation in the following.
Let I = (U,AT, τ) be a tolerance information system,
A ⊆ AT . For the fuzzy set X ∈ F (U), denote
RA(X)(x) = ∧{X(y) | y ∈ RA(x)},RA(X)(x) = ∨{X(y) | y ∈ RA(x)}, (3)
where “ ∨ ” means “max” and “ ∧ ” means “min”, then
RA(X) and RA(X) are the lower and upper approximation
of the fuzzy set X over the tolerance relation R with respect
to the subset of attributes A. If RA(X) �= RA(X), then the
fuzzy set X is a fuzzy rough set on the tolerance relation.
We can easily find that this model will be the fuzzy rough
set model we have introduced if the above relation R is an
equivalence relation.
A. The First Type Of Multi-granulation Fuzzy Rough Set OnTolerance Relations
First, the first type of two-granulation fuzzy rough set (in
brief 1st TGFRS) on tolerance relations of a fuzzy set is
defined.
Definition 3.1: Let I = (U,AT, τ) be a tolerance infor-
mation system, A,B ⊆ AT . For the fuzzy set X ∈ F (U),denote
FRA+B(X)(x) = {∧{X(y) | y ∈ RA(x)}}∨{∧{X(y) | y ∈ RB(x)}},
FRA+B(X)(x) = {∨{X(y) | y ∈ RA(x)}}∧{∨{X(y) | y ∈ RB(x)}},
(4)
where “ ∨ ” means “max” and “ ∧ ” means “min”, then
FRA+B(X) and FRA+B(X) are respectively called the
first type of two-granulation lower approximation and up-
per approximation of X on tolerance relations R with
respect to the subsets of attributes A and B. X is a two-
granulation fuzzy rough set on tolerance relations if and
358
TABLE I
A FUZZY TARGET INFORMATION SYSTEM
U a1 a2 a3 dx1 2 1 3 0.6x2 3 2 1 0.7x3 2 3 3 0.7x4 2 2 3 0.9x5 1 1 4 0.5x6 1 3 2 0.4x7 3 2 1 0.7x8 1 1 4 0.5x9 2 1 2 0.8x10 3 1 2 0.7
only if FRA+B(X) �= FRA+B(X). Otherwise, X is a two-
granulation fuzzy definable set on tolerance relations. The
boundary of the set X is defined as
BndFRA+B
(X) = FRA+B(X) ∩ (∼ FRA+B(X)). (5)
It can be found that the 1st TGFRS on tolerance relations
will be degenerated into fuzzy rough set when A = B and
RA(x) and RB(x) are equivalence classes with respect to the
subsets of attributes A and B. That is to say, a fuzzy rough
set model is a special instance of the 1st TGFRS on tolerance
relations. What’s more, the 1st TGFRS on tolerance relations
will be degenerated into a rough set model on a tolerance
relation if A = B and the considered concept X is a crisp
set.
In the following, we employ an example to illustrate the
above concepts.
Example 3.1: A fuzzy target information system are given
in Table I. The universe U = {x1, x2, · · · , x10}, the set of
condition attributes AT = {a1, a2, a3}, the set of decision
attribute D = {d}. If suppose A1 = {a1, a2} and A2 ={a1, a3}, we consider the first type of two-granulation lower
and upper approximation of D with respect to A1 and A2,
where the tolerance relation is defined as RAi= {(xi, yj) ∈
U × U | |f(xi, a) − f(xj , a)| ≤ 1, a ∈ Ai}. For the fuzzy
set
D = {0.6, 0.7, 0.7, 0.9, 0.5, 0.4, 0.7, 0.7, 0.8, 0.7},the single granulation lower and upper approximation on a
tolerance relation are
RA1(D) = {0.5, 0.6, 0.4, 0.4, 0.5, 0.4, 0.6, 0.5, 0.5, 0.6},RA1(D) = {0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9};RA2(D) = {0.4, 0.7, 0.4, 0.4, 0.5, 0.4, 0.7, 0.5, 0.4, 0.6},RA2(D) = {0.9, 0.8, 0.9, 0.9, 0.9, 0.9, 0.8, 0.9, 0.9, 0.9};
RA1∪A2(D) = {0.5, 0.7, 0.4, 0.4, 0.5, 0.4, 0.7, 0.5, 0.6, 0.6},RA1∪A2(D) = {0.9, 0.8, 0.9, 0.9, 0.9, 0.9, 0.8, 0.9, 0.9, 0.9}.From Definition 3.1, we can compute the first type of two-
granulation lower and upper approximation of D on tolerance
relations are
FRA1+A2(D) ={0.5, 0.7, 0.4, 0.4, 0.5, 0.4, 0.7, 0.5, 0.5, 0.6},FRA1+A2(D) ={0.9, 0.8, 0.9, 0.9, 0.9, 0.9, 0.8, 0.9, 0.9, 0.9}.
Obviously, the following can be found
FRA1+A2(D) = RA1(D) ∪ RA2(D),
FRA1+A2(D) = RA1(D) ∩ RA2(D),FRA1+A2(D) ⊆ RA1∪A2(D) ⊆ D
⊆ RA1∪A2(D) ⊆ FRA1+A2(D).
Just from Definition 3.1, we can obtain some properties
of the 1st TGFRS in a tolerance information system.Proposition 3.1: Let I = (U,AT, τ) be a tolerance
information system, B,A ⊆ AT and X ∈ F (U). Then the
following properties hold.(1) FRA+B(X) ⊆ X,
(2) FRA+B(X) ⊇ X;(3) FRA+B(∼ X) =∼ FRA+B(X),(4) FRA+B(∼ X) =∼ FRA+B(X);(5) FRA+B(U) = FRA+B(U) = U,
(6) FRA+B(∅) = FRA+B(∅) = ∅;(7) FRA+B(X) ⊇ FRA+B(FRA+B(X)),(8) FRA+B(X) ⊆ FRA+B(FRA+B(X)).Proof: We only prove (1),(2) and (3) and the rest can been
proved by Definition 3.1. It is obvious that all terms hold
when A = B. When A �= B, the proposition can be proved
as follows.(1) For any x ∈ U and A, B ⊆ AT , since RA(X) ⊆ X
and RB(X) ⊆ X , we know
∧{X(y) | y ∈ RA(x)} ≤ X(y)
and
∧{X(y) | y ∈ RB(x)} ≤ X(y)
Therefore,
{∧{X(y) | y ∈ RA(x)}}∨{∧{X(y) | y ∈ RB(x)}} ≤ X(y).
i.e., FRA+B(X) ⊆ X.
(2) For any x ∈ U and A, B ⊆ AT , since X ⊆ RA(X)and X ⊆ RA(X), we know
X(y) ≤ ∨{X(y) | y ∈ RA(x)}and
X(y) ≤ ∨{X(y) | y ∈ RB(x)}Therefore,
X(y) ≤ {∨{X(y) | y ∈ RA(x)}}∧{∨{X(y) | y ∈ RB(x)}}.i.e., X ⊆ FRA+B(X).
(3) For any x ∈ U and A, B ⊆ AT , since RA(∼ X) =∼RA(X) and RB(∼ X) =∼ RB(X), then we have
FRA+B(∼ X)
= {∧{1 − X(y) | y ∈ RA(x)}}∨ {∧{1 − X(y) | y ∈ RB(x)}}
= {1 − ∨{X(y) | y ∈ RA(x)}}∨ {1 − ∨{X(y) | y ∈ RB(x)}}
= 1 − {∨{X(y) | y ∈ RA(x)}}∧ {∨{X(y) | y ∈ RB(x)}}
=∼ FRA+B(X).
359
Proposition 3.2: Let I = (U,AT, F ) be an information
system, B,A ⊆ AT, X, Y ∈ F (U). Then the following
properties hold.
(1) FRA+B(X ∩ Y ) ⊆ FRA+B(X) ∩ FRA+B(Y ),(2) FRA+B(X ∪ Y ) ⊇ FRA+B(X) ∪ FRA+B(Y );(3) X ⊆ Y ⇒ FRA+B(X) ⊆ FRA+B(Y ),(4) X ⊆ Y ⇒ FRA+B(X) ⊆ FRA+B(Y );(5) FRA+B(X ∪ Y ) ⊇ FRA+B(X) ∪ FRA+B(Y ),(6) FRA+B(X ∩ Y ) ⊆ FRA+B(X) ∩ FRA+B(Y ).
According to the similarity of the properties, we
only prove the odd items. All terms hold obviously when
A = B or X = Y . If A �= B and X �= Y , the proposition
can be proved as follows.
(1) For any x ∈ U , A,B ⊆ AT and X, Y ∈ F (U),
FRA+B(X ∩ Y )(x)
= {∧{(X ∩ Y )(y) | y ∈ RA(x)}}∨ {∧{(X ∩ Y )(y) | y ∈ RB(x)}}
= {∧{X(y) ∧ Y (y) | y ∈ RA(x)}}∨ {∧{X(y) ∧ Y (y) | y ∈ RB(x)}}
= {RA(X)(x) ∧ RA(Y )(x)} ∨ {RB(X)(x) ∧ RB(Y )(x)}≤ {RA(X)(x) ∨ RB(X)(x)} ∧ {RA(Y )(x) ∨ RB(Y )(x)}= FRA+B(X)(x) ∧ FRA+B(Y )(x).
Then FRA+B(X ∩ Y ) ⊆ FRA+B(X) ∩ FRA+B(Y ).(3) Since for any x ∈ U , we have X(y) ≤ Y (y). Then
the properties hold obviously by Definition 3.1.
(5) Since X ⊆ X ∪ Y , and Y ⊆ X ∪ Y , then
FRA+B(X) ⊆ FRA+B(X ∪ Y ) and FRA+B(Y ) ⊆FRA+B(X ∪ Y ). So the property FRA+B(X ∪ Y ) ⊇FRA+B(X) ∪ FRA+B(Y ) obviously holds.
The proposition was proved.
The lower and upper approximation in Definition 3.1 are
a pair of fuzzy sets. If we associate the cut set of a fuzzy set,
we can make a description of a fuzzy set X by a classical
set in an information system.
Definition 3.2: Let I = (U,AT, τ) be a tolerance in-
formation system, A,B ⊆ AT and X ⊆ U . For any
0 < β ≤ α ≤ 1, the lower approximation FRA+B(X)and upper approximation FRA+B(X) of X about the α,
β cut sets on tolerance relations RA and RB are defined,
respectively, as follows
FRA+B(X)α = {x ∈ U | FRA+B(X)(x) ≥ α},FRA+B(X)β = {x ∈ U | FRA+B(X)(x) ≥ β}. (6)
FRA+B(X)α can be explained as the set of objects in
U which surely belong to X on tolerance relations RA and
RB and the memberships of which are more than α, while
FRA+B(X)β is the set of objects in U which possibly
belong to X on tolerance relations RA and RB and the
memberships of which are more than β.
Proposition 3.3: Let I = (U,AT, τ) be a tolerance
information system, A,B ⊆ AT and X, Y ⊆ U . For any
0 < β ≤ α ≤ 1, we have
(1) FRA+B(X ∩ Y )α ⊆ FRA+B(X)α ∩ FRA+B(Y )α,
(2) FRA+B(X ∪ Y )β ⊇ FRA+B(X)β ∪ FRA+B(Y )β ;(3) X ⊆ Y ⇒ FRA+B(X)α ⊆ FRA+B(Y )α,
(4) X ⊆ Y ⇒ FRA+B(X)β ⊆ FRA+B(Y )β ;(5) FRA+B(X ∪ Y )α ⊇ FRA+B(X)α ∪ FRA+B(Y )α,
(6) FRA+B(X ∩ Y )β ⊆ FRA+B(X)β ∩ FRA+B(Y )β .It is easy to prove by Definition 3.2 and Proposition
3.2.
In the following, we will introduce the first type of
multi-granulation fuzzy rough set (in brief 1st MGFRS) on
tolerance relations and its corresponding properties by ex-
tending the 1st two-granulation fuzzy rough set on tolerance
relations.
Definition 3.3: Let I = (U,AT, τ) be a tolerance infor-
mation system, Ai ⊆ AT, i = 1, · · · ,m . For the fuzzy set
X ∈ F (U), denote
FR m∑
i=1Ai
(X)(x) =m∨
i=1
{∧{X(y) | y ∈ RAi(x)}},
FR m∑
i=1Ai
(X)(x) =m∧
i=1
{∨{X(y) | y ∈ RAi(x)}},
(7)
where “∨
” means “max” and “∧
” means “min”, then
FR m∑
i=1Ai
(X) and FR m∑
i=1Ai
(X) are respectively called the
first type of multi-granulation lower approximation and upper
approximation of X on the tolerance relations RAi(i =
1, · · · ,m). X is a multi-granulation fuzzy rough set on
the tolerance relations RAi(i = 1, · · · ,m) if and only if
FR m∑
i=1Ai
(X) �= FR m∑
i=1Ai
(X). Otherwise, X is a multi-
granulation fuzzy definable set on the tolerance relations
RAi(i = 1, · · · ,m). The boundary of the set X is defined
as
BndFR m∑
i=1Ai
(X) = FR m∑
i=1Ai
(X) ∩ (∼ FR m∑
i=1Ai
(X)).
(8)
It can be found that the 1st MGFRS on the tolerance
relations RAi(i = 1, · · · ,m) will be degenerated into fuzzy
rough set when Ai = Aj , i �= j and RAi(x) are equivalence
classes with respect to the subsets of attributes Ai(i =1, · · · ,m). That is to say, a fuzzy rough set model is a
special instance of the 1st MGFRS on the tolerance relations.
What’s more, the 1st MGFRS on tolerance relations will be
degenerated into a rough set model on tolerance relation if
Ai = Aj , i �= j and the considered concept X is a crisp set.
The properties about 1st MGFRS on tolerance relations
are listed in the following which can be extended from the
1st TGFRS model on tolerance relations.
Proposition 3.4: Let I = (U,AT, τ) be a tolerance infor-
mation system, Ai ⊆ AT, i = 1, · · · ,m and X ∈ F (U).Then the following properties hold.
(1) FR m∑
i=1Ai
(X) ⊆ X,
(2) FR m∑
i=1Ai
(X) ⊇ X;
(3) FR m∑
i=1Ai
(∼ X) =∼ R m∑
i=1Ai
(X),
Proof:
Proof:
360
(4) FR m∑
i=1Ai
(∼ X) =∼ FR m∑
i=1Ai
(X);
(5) FR m∑
i=1Ai
(U) = FR m∑
i=1Ai
(U) = U,
(6) FR m∑
i=1Ai
(∅) = FR m∑
i=1Ai
(∅) = ∅;
(7) FR m∑
i=1Ai
(X) ⊇ FR m∑
i=1Ai
(FR m∑
i=1Ai
(X)),
(8) FR m∑
i=1Ai
(X) ⊆ FR m∑
i=1Ai
(FR m∑
i=1Ai
(X)).
The proof is similar to Proposition 3.1.
Proposition 3.5: Let I = (U,AT, τ) be a tolerance
information system, Ai ⊆ AT, i = 1, · · · ,m, X, Y ∈ F (U).Then the following properties hold.
(1) FR m∑
i=1Ai
(X ∩ Y ) ⊆ FR m∑
i=1Ai
(X) ∩ FR m∑
i=1Ai
(Y ),
(2) FR m∑
i=1Ai
(X ∪ Y ) ⊇ FR m∑
i=1Ai
(X) ∪ FR m∑
i=1Ai
(Y );
(3) X ⊆ Y ⇒ FR m∑
i=1Ai
(X) ⊆ FR m∑
i=1Ai
(Y ),
(4) X ⊆ Y ⇒ FR m∑
i=1Ai
(X) ⊆ FR m∑
i=1Ai
(Y );
(5) FR m∑
i=1Ai
(X ∪ Y ) ⊇ FR m∑
i=1Ai
(X) ∪ FR m∑
i=1Ai
(Y ),
(6) FR m∑
i=1Ai
(X ∩ Y ) ⊆ FR m∑
i=1Ai
(X) ∩ FR m∑
i=1Ai
(Y ).
proof The proof of this proposition is similar to Proposi-
tion 3.2.
Definition 3.4: Let I = (U,AT, τ) be a tolerance infor-
mation system, Ai ⊆ AT, 1 ≤ i ≤ m, and X ⊆ U . For
any 0 < β ≤ α ≤ 1, the lower approximation FR m∑
i=1Ai
(X)
and upper approximation FR m∑
i=1Ai
(X) of X about the α, β
cut sets on the tolerance relations RAi(i = 1, · · · ,m) are
defined, respectively, as follows
FR m∑
i=1Ai
(X)α = {x ∈ U | FR m∑
i=1Ai
(X)(x) ≥ α},
FR m∑
i=1Ai
(X)β = {x ∈ U | FR m∑
i=1Ai
(X)(x) ≥ β}.(9)
FR m∑
i=1Ai
(X)α can be explained as the set of objects in U
which surely belong to X on the tolerance relations RAi(i =
1, · · · ,m) and the memberships of which are more than α,
while FR m∑
i=1Ai
(X)β is the set of objects in U which possibly
belong to X on the tolerance relations RAi(i = 1, · · · ,m)
and the memberships of which are more than β.
Proposition 3.6: Let I = (U,AT, τ) be a tolerance infor-
mation system, Ai ⊆ AT, i = 1, · · · ,m, and X, Y ⊆ U .
For any 0 < β ≤ α ≤ 1, we have
(1) FR m∑
i=1Ai
(X∩Y )α ⊆ FR m∑
i=1Ai
(X)α∩FR m∑
i=1Ai
(Y )α,
(2) FR m∑
i=1Ai
(X ∪Y )β ⊇ FR m∑
i=1Ai
(X)β ∪FR m∑
i=1Ai
(Y )β ;
(3) X ⊆ Y ⇒ FR m∑
i=1Ai
(X)α ⊆ FR m∑
i=1Ai
(Y )α,
(4) X ⊆ Y ⇒ FR m∑
i=1Ai
(X)β ⊆ FR m∑
i=1Ai
(Y )β ;
(5) FR m∑
i=1Ai
(X∪Y )α ⊇ FR m∑
i=1Ai
(X)α∪FR m∑
i=1Ai
(Y )α,
(6) FR m∑
i=1Ai
(X ∩Y )β ⊆ FR m∑
i=1Ai
(X)β ∩FR m∑
i=1Ai
(Y )β .
It is easy to prove by Definition 3.4 and Proposition
3.5.
B. The Second Type Of Multi-granulation Fuzzy Rough SetOn Tolerance Relations
In this subsection, we will propose another type of
MGFRS on tolerance relations. We first define the sec-
ond type of two-granulation fuzzy rough set (in brief 2nd
TGFRS)on tolerance relations.
Definition 3.5: Let I = (U,AT, τ) be a tolerance infor-
mation system, A,B ⊆ AT . For the fuzzy set X ∈ F (U),denote
SRA+B(X)(x) = {∧{X(y) | y ∈ RA(x)}}∧{∧{X(y) | y ∈ RB(x)}},
SRA+B(X)(x) = {∨{X(y) | y ∈ RA(x)}}∨{∨{X(y) | y ∈ RB(x)}},
(10)
then SRA+B(X) and SRA+B(X) are respectively called
the second type of two-granulation lower approximation and
upper approximation of X on the tolerance relations Rwith respect to the subsets of attributes A and B. X is
the second type of two-granulation fuzzy rough set on the
tolerance relations if and only if SRA+B(X) �= SRA+B(X).Otherwise, X is the second type of two-granulation fuzzy
definable set on the tolerance relations. The boundary of the
set X is defined as
BndSRA+B
(X) = SRA+B(X) ∩ (∼ SRA+B(X)). (11)
It can be found that the 2nd TGFRS on tolerance relations
will be degenerated into the fuzzy rough set model when
A = B and RA(x) and RB(x) are equivalence classes with
respect to the subsets of attributes A and B. That is, a fuzzy
rough set model is a special instance of the 2nd TGFRS
on tolerance relations. What’s more, the 2nd TGFRS on
tolerance relations will be degenerated into a rough set model
on a tolerance relation if A = B and the considered concept
X is a crisp set.
In the following, we employ an example to illustrate the
above concepts.
Example 3.2: (Continued from Example 3.1) From Defini-
tion 3.2, we can compute the second type of two-granulation
lower and upper approximation of D on the tolerance relation
RA1 and RA2 are
SRA1+A2(D) ={0.4, 0.6, 0.4, 0.4, 0.5, 0.4, 0.6, 0.5, 0.4, 0.6},SRA1+A2(D) ={0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9}.
Proof:
Proof:
361
Obviously, the following can be found
SRA1+A2(D) = RA1(D) ∩ RA2(D),
SRA1+A2(D) = RA1(D) ∪ RA2(D),SRA1+A2(D) ⊆ RA1∪A2(D) ⊆ D
⊆ RA1∪A2(D) ⊆ SRA1+A2(D).
Proposition 3.7 Let I = (U,AT, τ) be a tolerance in-
formation system, B,A ⊆ AT and X ∈ F (U). Then the
following properties hold.
(1) SRA+B(X) ⊆ X,
(2) SRA+B(X) ⊇ X;(3) SRA+B(∼ X) =∼ SRA+B(X),
(4) SRA+B(∼ X) =∼ SRA+B(X);
(5) SRA+B(U) = SRA+B(U) = U,
(6) SRA+B(∅) = SRA+B(∅) = ∅;(7) SRA+B(X) ⊇ SRA+B(SRA+B(X)),
(8) SRA+B(X) ⊆ SRA+B(SRA+B(X)).The methods of the proof is similar to Proposition
3.1.
Proposition 3.8: Let I = (U,AT, τ) be a tolerance
information system, B,A ⊆ AT, X, Y ∈ F (U). Then the
following properties hold.
(1) SRA+B(X ∩ Y ) = SRA+B(X) ∩ SRA+B(Y ),
(2) SRA+B(X ∪ Y ) = SRA+B(X) ∪ SRA+B(Y );(3) X ⊆ Y ⇒ SRA+B(X) ⊆ SRA+B(Y ),
(4) X ⊆ Y ⇒ SRA+B(X) ⊆ SRA+B(Y );(5) SRA+B(X ∪ Y ) ⊇ SRA+B(X) ∪ SRA+B(Y ),
(6) SRA+B(X ∩ Y ) ⊆ SRA+B(X) ∩ SRA+B(Y ).Here we prove (1) and (2). All terms hold obviously
when A = B or X = Y . If A �= B and X �= Y , the
proposition can be proved as follows.
(1) For any x ∈ U , A,B ⊆ AT and X, Y ∈ F (U),
SRA+B(X ∩ Y )(x)
= {∧{(X ∩ Y )(y) | y ∈ RA(x)}}∧ {∧{(X ∩ Y )(y) | y ∈ RB(x)}}
= {∧{X(y) ∧ Y (y) | y ∈ RA(x)}}∧ {∧{X(y) ∧ Y (y) | y ∈ RB(x)}}
= {RA(X)(x) ∧ RA(Y )(x)} ∧ {RB(X)(x) ∧ RB(Y )(x)}= {RA(X)(x) ∧ RB(X)(x)} ∧ {RA(Y )(x) ∧ RB(Y )(x)}= RA+B(X)(x) ∧ RA+B(Y )(x).
Then SRA+B(X ∩ Y ) = SRA+B(X) ∩ SRA+B(Y ).(2) Similarly, for any x ∈ U , A,B ⊆ AT and X, Y ∈
F (U),
SRA+B(X ∪ Y )(x)= {∨{(X ∪ Y )(y) | y ∈ RA(x)}}∨ {∨{(X ∪ Y )(y) | y ∈ RB(x)}}
= {∨{X(y) ∨ Y (y) | y ∈ RA(x)}}∨ {∨{X(y) ∨ Y (y) | y ∈ RB(x)}}
= {RA(X)(x) ∨ RA(Y )(x)} ∨ {RB(X)(x) ∨ RB(Y )(x)}= {RA(X)(x) ∨ RB(X)(x)} ∨ {RA(Y )(x) ∨ RB(Y )(x)}= SRA+B(X)(x) ∨ SRA+B(Y )(x).
Then SRA+B(X ∪ Y ) = SRA+B(X) ∪ SRA+B(Y ).Definition 3.6: Let I = (U,AT, τ) be a tolerance in-
formation system, A,B ⊆ AT and X ⊆ U . For any
0 < β ≤ α ≤ 1, the lower approximation SRA+B(X)and upper approximation SRA+B(X) of X about the α,
β cut sets on tolerance relations RA and RB are defined,
respectively, as follows
SRA+B(X)α = {x ∈ U | SRA+B(X)(x) ≥ α},SRA+B(X)β = {x ∈ U | SRA+B(X)(x) ≥ β}.
(12)
SRA+B(X)α can be explained as the set of objects in
U which surely belong to X on tolerance relations RA and
RB and the memberships of which are more than α, while
SRA+B(X)β is the set of objects in U which possibly belong
to X on tolerance relations RA and RB and the memberships
of which are more than β.
Proposition 3.9: Let I = (U,AT, τ) be a tolerance
information system, A,B ⊆ AT and X, Y ⊆ U . For any
0 < β ≤ α ≤ 1, we have
(1) SRA+B(X ∩ Y )α = SRA+B(X)α ∩ SRA+B(Y )α,
(2) SRA+B(X ∪ Y )β = SRA+B(X)β ∪ SRA+B(Y )β ;(3) X ⊆ Y ⇒ SRA+B(X)α ⊆ SRA+B(Y )α,
(4) X ⊆ Y ⇒ SRA+B(X)β ⊆ SRA+B(Y )β ;(5) SRA+B(X ∪ Y )α ⊇ SRA+B(X)α ∪ SRA+B(Y )α,
(6) SRA+B(X ∩ Y )β ⊆ SRA+B(X)β ∩ SRA+B(Y )β .It is easy to prove by Definition 3.6 and Proposition
3.8.
In the following, we will introduce the second type of
multi-granulation fuzzy rough set (in brief 2nd MGFRS)
on tolerance relations and its corresponding properties by
extending the second type of two-granulation fuzzy rough
set on tolerance relations.
Definition 3.7: Let I = (U,AT, τ) be a tolerance infor-
mation system, Ai ⊆ AT, i = 1, · · · ,m. For the fuzzy set
X ∈ F (U), denote
SR m∑
i=1Ai
(X)(x) =m∧
i=1
{∧{X(y) | y ∈ RAi(x)}},
SR m∑
i=1Ai
(X)(x) =m∨
i=1
{∨{X(y) | y ∈ RAi(x)}},
(13)
where “∨
” means “max” and “∧
” means “min”, then
SR m∑
i=1Ai
(X) and SR m∑
i=1Ai
(X) are respectively called the
Proof:
Proof:Proof:
362
second type of multi-granulation lower approximation and
upper approximation of X on tolerance relations R with
respect to the subsets of attributes Ai(i = 1, · · · ,m). Xis the second type of multi-granulation fuzzy rough set if
and only if SR m∑
i=1Ai
(X) �= SR m∑
i=1Ai
(X). Otherwise, X is
the second type of multi-granulation fuzzy definable set on
tolerance relations. The boundary of the set X is defined as
BndSR m∑
i=1Ai
(X) = SR m∑
i=1Ai
(X) ∩ (∼ SR m∑
i=1Ai
(X)).
(14)
It can be found that the 2nd MGFRS will be degenerated
into fuzzy rough set when Ai = Aj , i �= j and RAi(x) are
equivalence classes with respect to the subsets of attributes
Ai(i = 1, · · · ,m). That is , a fuzzy rough set model is also
a special instance of the 2nd MGFRS on tolerance relations.
What’s more, the MGFRS 2nd will be degenerated into a
rough set model on a tolerance relation if Ai = Aj , i �= jand the considered concept X is a crisp set.
The properties about the 2nd MGFRS on tolerance rela-
tions are listed in the following which can be extended from
the 2nd TGFRS model on tolerance relations.
Proposition 3.10: Let I = (U,AT, τ) be a tolerance infor-
mation system, Ai ⊆ AT, 1 ≤ i ≤ m and X ∈ F (U). Then
the following properties hold.
(1) SR m∑
i=1Ai
(X) ⊆ X,
(2) SR m∑
i=1Ai
(X) ⊇ X;
(3) SR m∑
i=1Ai
(∼ X) =∼ SR m∑
i=1Ai
(X),
(4) SR m∑
i=1Ai
(∼ X) =∼ SR m∑
i=1Ai
(X);
(5) SR m∑
i=1Ai
(U) = SR m∑
i=1Ai
(U) = U,
(6) SR m∑
i=1Ai
(∅) = SR m∑
i=1Ai
(∅) = ∅.
(7) SR m∑
i=1Ai
(X) ⊇ SR m∑
i=1Ai
(SRA+B(X)),
(8) SR m∑
i=1Ai
(X) ⊆ SR m∑
i=1Ai
(SRA+B(X)).
The proof is similar to Proposition 3.7.
Proposition 3.11: Let I = (U,AT, τ) be a tolerance
information system, Ai ⊆ AT, 1 ≤ i ≤ m, X, Y ∈ F (U).Then the following properties hold.
(1) SR m∑
i=1Ai
(X ∩ Y ) = SR m∑
i=1Ai
(X) ∩ SR m∑
i=1Ai
(Y ),
(2) SR m∑
i=1Ai
(X ∪ Y ) = SR m∑
i=1Ai
(X) ∪ SR m∑
i=1Ai
(Y );
(3) X ⊆ Y ⇒ SR m∑
i=1Ai
(X) ⊆ SR m∑
i=1Ai
(Y ),
(4) X ⊆ Y ⇒ SR m∑
i=1Ai
(X) ⊆ SR m∑
i=1Ai
(Y );
(5) SR m∑
i=1Ai
(X ∪ Y ) ⊇ SR m∑
i=1Ai
(X) ∪ SR m∑
i=1Ai
(Y ),
(6) SR m∑
i=1Ai
(X ∩ Y ) ⊆ SR m∑
i=1Ai
(X) ∩ SR m∑
i=1Ai
(Y ).
The proof of this proposition is similar to Proposi-
tion 3.8.Definition 3.8: Let I = (U,AT, τ) be a tolerance infor-
mation system, Ai ⊆ AT, 1 ≤ i ≤ m, and X ⊆ U . For any
0 < β ≤ α ≤ 1, the lower approximation SR m∑
i=1Ai
(X) and
upper approximation SR m∑
i=1Ai
(X) of X about the α, β cut
sets on tolerance relations RAi(i = 1, · · · ,m) are defined,
respectively, as follows
SR m∑
i=1Ai
(X)α = {x ∈ U | SR m∑
i=1Ai
(X)(x) ≥ α},
SR m∑
i=1Ai
(X)β = {x ∈ U | SR m∑
i=1Ai
(X)(x) ≥ β}.(15)
SR m∑
i=1Ai
(X)α can be explained as the set of objects in
U which surely belong to X on tolerance relations RAi(i =1, · · · ,m) and the memberships of which are more than α,
while SR m∑
i=1Ai
(X)β is the set of objects in U which possibly
belong to X on tolerance relations RAi(i = 1, · · · ,m) and
the memberships of which are more than β.Proposition 3.12: Let I = (U,AT, τ) be a tolerance
information system, Ai ⊆ AT, 1 ≤ i ≤ m, and X, Y ⊆ U .
For any 0 < β ≤ α ≤ 1, we have(1) SR m∑
i=1Ai
(X ∩ Y )α = SR m∑
i=1Ai
(X)α ∩ SR m∑
i=1Ai
(Y )α,
(2) SR m∑
i=1Ai
(X ∪ Y )β = SR m∑
i=1Ai
(X)β ∪ SR m∑
i=1Ai
(Y )β ;
(3) X ⊆ Y ⇒ SR m∑
i=1Ai
(X)α ⊆ SR m∑
i=1Ai
(Y )α,
(4) X ⊆ Y ⇒ SR m∑
i=1Ai
(X)β ⊆ SR m∑
i=1Ai
(Y )β ;
(5) SR m∑
i=1Ai
(X ∪ Y )α ⊇ SR m∑
i=1Ai
(X)α ∪ SR m∑
i=1Ai
(Y )α,
(6) SR m∑
i=1Ai
(X ∩ Y )β ⊆ SR m∑
i=1Ai
(X)β ∩ SR m∑
i=1Ai
(Y )β .
It is easy to prove by Definition 3.8 and Proposition
3.11.On the basis of tolerance relations, we will investigate the
interrelationship among SGFRS, the 1st MGFRS and the 2nd
MGFRS in this section after the discussion of the properties
of them.Proposition 3.13: Let I = (U,AT, τ) be a tolerance
information system, Ai ⊆ AT, 1 ≤ i ≤ m, X ∈ F (U).Then the following properties hold.
(1) FR m∑
i=1Ai
(X) =m⋃
i=1
RAi(X),
(2) FR m∑
i=1Ai
(X) =m⋂
i=1
RAi(X);
(3) SR m∑
i=1Ai
(X) =m⋂
i=1
RAi(X),
(4) SR m∑
i=1Ai
(X) =m⋃
i=1
RAi(X);
(5) SR m∑
i=1Ai
(X) ⊆ FR m∑
i=1Ai
(X) ⊆ R m⋃
i=1Ai
(X);
Proof:
Proof:
Proof:
363
(6) SR m∑
i=1Ai
(X) ⊇ FR m∑
i=1Ai
(X) ⊇ R m⋃
i=1Ai
(X).
(7) SR m∑
i=1Ai
(X) ⊆ RAi(X) ⊆ FR m∑
i=1Ai
(X);
(8) SR m∑
i=1Ai
(X) ⊇ RAi(X) ⊇ FR m∑
i=1Ai
(X).
The proof can be obtained by Definition 3.3, 3.7.
IV. CONCLUSIONS
The theories of rough sets and fuzzy sets both extended
the classical set theory in terms of dealing with uncertainty
and imprecision. However, the theory of fuzzy set pay more
attention to the fuzziness of knowledge while the theory of
rough set to the roughness of knowledge in the point view of
granular of knowledge. For the complement of the two types
of theory, fuzzy rough set models are investigated to solve
practical problems. Given that the equivalence relations in the
fuzzy rough set theory is too rigorous for some practical ap-
plication, it is necessary to weaken the equivalence relations
to tolerance relations. The contribution of this paper is having
constructed two new types of fuzzy rough sets on tolerance
relations associated with granular computing called multi-
granulation fuzzy rough set models on tolerance relations,
in which the set approximation operators are defined on the
basis of multiple tolerance relations. What’s more, we make
conclusions that fuzzy rough set model and rough set model
on a tolerance relation are special cases of the two types of
multi-granulation fuzzy rough set on tolerance relations by
analyzing the definition of them. More properties of the two
types of fuzzy rough set on tolerance relations are discussed
and comparison are made with fuzzy rough set on a tolerance
relation. The construction of the new types of fuzzy rough
set models on tolerance relations is an extension in the point
view of granular computing and is meaningful in terms of
the generalization of rough set theory. We will investigate
the knowledge reduction in depth.
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Proof:
364
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